Properties

Label 12.22.a.a
Level 12
Weight 22
Character orbit 12.a
Self dual Yes
Analytic conductor 33.537
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 22 \)
Character orbit: \([\chi]\) = 12.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(33.5372813144\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 59049q^{3} \) \(\mathstrut -\mathstrut 11268090q^{5} \) \(\mathstrut +\mathstrut 281914136q^{7} \) \(\mathstrut +\mathstrut 3486784401q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 59049q^{3} \) \(\mathstrut -\mathstrut 11268090q^{5} \) \(\mathstrut +\mathstrut 281914136q^{7} \) \(\mathstrut +\mathstrut 3486784401q^{9} \) \(\mathstrut -\mathstrut 36172082484q^{11} \) \(\mathstrut -\mathstrut 449098578370q^{13} \) \(\mathstrut -\mathstrut 665369446410q^{15} \) \(\mathstrut +\mathstrut 2121858786546q^{17} \) \(\mathstrut -\mathstrut 4609406233900q^{19} \) \(\mathstrut +\mathstrut 16646747816664q^{21} \) \(\mathstrut +\mathstrut 95095276921656q^{23} \) \(\mathstrut -\mathstrut 349867305955025q^{25} \) \(\mathstrut +\mathstrut 205891132094649q^{27} \) \(\mathstrut -\mathstrut 2245742383351266q^{29} \) \(\mathstrut -\mathstrut 3155693201792656q^{31} \) \(\mathstrut -\mathstrut 2135925298597716q^{33} \) \(\mathstrut -\mathstrut 3176633856720240q^{35} \) \(\mathstrut -\mathstrut 18178503074861482q^{37} \) \(\mathstrut -\mathstrut 26518821954170130q^{39} \) \(\mathstrut -\mathstrut 169649739387485910q^{41} \) \(\mathstrut -\mathstrut 158968551608988244q^{43} \) \(\mathstrut -\mathstrut 39289400441064090q^{45} \) \(\mathstrut -\mathstrut 134697468442682736q^{47} \) \(\mathstrut -\mathstrut 479070284006657511q^{49} \) \(\mathstrut +\mathstrut 125293639486754754q^{51} \) \(\mathstrut -\mathstrut 15637375269722538q^{53} \) \(\mathstrut +\mathstrut 407590280917135560q^{55} \) \(\mathstrut -\mathstrut 272180828705561100q^{57} \) \(\mathstrut +\mathstrut 2977241337691499484q^{59} \) \(\mathstrut +\mathstrut 3603855625679330702q^{61} \) \(\mathstrut +\mathstrut 982973811826192536q^{63} \) \(\mathstrut +\mathstrut 5060483199945213300q^{65} \) \(\mathstrut +\mathstrut 21066199531967164004q^{67} \) \(\mathstrut +\mathstrut 5615281006946865144q^{69} \) \(\mathstrut +\mathstrut 21980089544074358760q^{71} \) \(\mathstrut -\mathstrut 17054415965500339222q^{73} \) \(\mathstrut -\mathstrut 20659314549338271225q^{75} \) \(\mathstrut -\mathstrut 10197421380797593824q^{77} \) \(\mathstrut -\mathstrut 115020124425041803552q^{79} \) \(\mathstrut +\mathstrut 12157665459056928801q^{81} \) \(\mathstrut -\mathstrut 96628520442403345644q^{83} \) \(\mathstrut -\mathstrut 23909295774091117140q^{85} \) \(\mathstrut -\mathstrut 132608841994508906034q^{87} \) \(\mathstrut +\mathstrut 60427571095732966650q^{89} \) \(\mathstrut -\mathstrut 126607237700006838320q^{91} \) \(\mathstrut -\mathstrut 186340527872654544144q^{93} \) \(\mathstrut +\mathstrut 51939204290146251000q^{95} \) \(\mathstrut -\mathstrut 407820224794143352798q^{97} \) \(\mathstrut -\mathstrut 126124252956896532084q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 59049.0 0 −1.12681e7 0 2.81914e8 0 3.48678e9 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5} \) \(\mathstrut +\mathstrut 11268090 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(12))\).